334 XV NORMED ALGEBRAS AND SPECTRAL THEORY 4. BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS Let A be an algebra over the field C of complex numbers. An involution on A is a bijection x*-+x* of A onto A, satisfying the following conditions: (15.4.1) (x*)* = x, (x + y)* = x* + /", (Ax)* = Ix*, (xj;)* = y*x* for all x, y in A and all A e C. The element x* is often called the adjoint of x, and a subset of A which is stable under the involution is said to be selfi adjoint. If A has a unit element e, then x - e* = (e - x*)* = x** = x, and similarly e* . jc = x for all x 6 A, by virtue of (15.4.1). Hence (15.4.2) e* = e. If x is invertible in A, then (x"""1)*** = (xx""1)* = e* = e, and similarly x^x'1)* = e> by (15.4.1) and (15.4.2). Hence x* is invertible, and (15.4.3) (x-1)* = (x*)"1. A normed algebra with involution is a normed algebra A equipped with an involution xi—>x* such that (15.4.4) ||x*|| = ||x|| for all x e A. A star algebra is a Banach algebra A equipped with an involution xh-»x* such that (15.4.5) ||x||2 = ||x*x|| for all x 6 A. A star algebra satisfies (15.4.4), for it follows from (15.4.5) and (15.1.1) that ||x||2 g ||x|| • ||x*||, whence ||x|| ^ ||x*|| for all xe A; interchanging x and x*, we obtain (15.4.4). Examples of Banach Algebras with Involution and Star Algebras (15.4.6) If X is a metrizable compact space, the Banach algebra ^C(X) is a star algebra with respect to the involution/W/ (the complex conjugate of/). If X consists of a single point, #C(X) is isomorphic to the field C.itely differentiable complex-valued functions on [0, 1].